U.S. patent application number 13/803536 was filed with the patent office on 2013-10-24 for self-constraint non-iterative grappa reconstruction with closed-form solution.
This patent application is currently assigned to THE OHIO STATE UNIVERSITY. The applicant listed for this patent is THE OHIO STATE UNIVERSITY. Invention is credited to Rizwan Ahmad, Yu Ding, Orlando Simonetti, Samuel Tze Luong Ting, Hui Xue.
Application Number | 20130278256 13/803536 |
Document ID | / |
Family ID | 49379513 |
Filed Date | 2013-10-24 |
United States Patent
Application |
20130278256 |
Kind Code |
A1 |
Ahmad; Rizwan ; et
al. |
October 24, 2013 |
SELF-CONSTRAINT NON-ITERATIVE GRAPPA RECONSTRUCTION WITH
CLOSED-FORM SOLUTION
Abstract
Parallel magnetic resonance imaging (pMRI) reconstruction
techniques are commonly used to reduce scan time by undersampling
the k-space data. In GRAPPA, a k-space based pMRI technique, the
missing k-space data are estimated by solving a set of linear
equations; however, this set of equations does not take advantage
of the correlations within the missing k-space data. All k-space
data in a neighborhood acquired from a phased-array coil are
correlated. The correlation can be estimated easily as a
self-constraint condition, and formulated as an extra set of linear
equations to improve the performance of GRAPPA. We propose a
modified k-space based pMRI technique call self-constraint GRAPPA
(SC-GRAPPA) which combines the linear equations of GRAPPA with
these extra equations to solve for the missing k-space data. Since
SC-GRAPPA utilizes a least-squares solution of the linear
equations, it has a closed-form solution that does not require an
iterative solver.
Inventors: |
Ahmad; Rizwan; (Columbus,
OH) ; Ding; Yu; (Columbus, OH) ; Simonetti;
Orlando; (Columbus, OH) ; Ting; Samuel Tze Luong;
(Columbus, OH) ; Xue; Hui; (Franklin Park,
NJ) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
THE OHIO STATE UNIVERSITY |
Columbus |
OH |
US |
|
|
Assignee: |
THE OHIO STATE UNIVERSITY
Columbus
OH
|
Family ID: |
49379513 |
Appl. No.: |
13/803536 |
Filed: |
March 14, 2013 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
61635410 |
Apr 19, 2012 |
|
|
|
Current U.S.
Class: |
324/309 |
Current CPC
Class: |
G01R 33/5611 20130101;
G01R 33/5612 20130101; G01R 33/4824 20130101 |
Class at
Publication: |
324/309 |
International
Class: |
G01R 33/561 20060101
G01R033/561; G01R 33/48 20060101 G01R033/48 |
Claims
1. A method of determining a Self-Constraint (SC) GeneRalized
Autocalibrating Partially Parallel Acquisition (GRAPPA)
reconstruction, comprising: sampling k-space data as a set of ACS
lines acquired from an MRI apparatus; estimating a GRAPPA kernel
from the ACS lines; estimating a null projection matrix N from the
ACS lines; performing a GRAPPA reconstruction to calculate k.sub.0;
and calculating a k-space of the SC-GRAPPA in accordance with the
GRAPPA reconstruction and a self-constraint condition defined by
local k-space correlations among acquired and unacquired k-space
samples.
2. The method of claim 1, further comprising estimating the GRAPPA
kernel using a linear regression.
3. The method of claim 1, further comprising estimated the null
projection matrix N using a parallel reconstruction using null
operations (PRUNO) method.
4. The method of claim 3, wherein the null projection matrix is
defined by: [ 0 0 ] = [ N 11 N 12 N 21 N 22 ] [ k acq k unacq ] ,
##EQU00010## wherein k.sub.unacq and k.sub.acq are vectorized
unacquired and acquired k-space data, respectively,
5. The method of claim 1, calculating a k-space of the SC-GRAPPA
further comprising solving a set of linear equations in accordance
with correlations within missing k-space data.
6. The method of claim 5, wherein the set of linear equations are
defined by: k=k.sub.0+w; and 0=Nk+v, wherein k.sub.0 represents the
GRAPPA reconstruction, w and v are random noise terms, and k is
vectorized local k-space data.
7. The method of claim 6, wherein calculating the k-space of the
SC-GRAPPA is performed in accordance with:
k=k.sub.0-RN.sup.T(Q+NRN.sup.T).sup.-1Nk.sub.0, where Q and R are
covariance matrices of w and v, respectively.
8. The method of claim 1, wherein the k-space of the SC-GRAPPA is
performed using a least squares problem with prior estimation.
9. The method of claim 8, wherein the prior estimation is
k.sub.0.
10. The method of claim 1, further comprising applying a channel
combination using either the sum-of-squares or B1-weighted
summation.
11. A tangible computer-readable medium having computer executable
instructions stored thereon that when executed by a processor of a
computing device determines a Self-Constraint (SC) GeneRalized
Autocalibrating Partially Parallel Acquisition (GRAPPA)
reconstruction, comprising: sampling k-space data, including a set
of ACS lines, from an MRI apparatus; estimating a GRAPPA kernel
from the ACS lines; estimating a null projection matrix N is
estimated from the ACS lines; performing a GRAPPA reconstruction to
calculate k.sub.0; and calculating a k-space of the SC-GRAPPA in
accordance with the GRAPPA reconstruction and a self-constraint
condition defined by local k-space correlations among acquired and
unacquired k-space samples.
12. The tangible computer-readable medium of claim 11, further
comprising estimated the null projection matrix N using a parallel
reconstruction using null operations (PRUNO) method.
13. The tangible computer-readable medium of claim 11, calculating
a k-space of the SC-GRAPPA further comprising solving a set of
linear equations in accordance with correlations within missing
k-space data.
14. The tangible computer-readable medium of claim 13, wherein the
set of linear equations are defined by: k=k.sub.0+w; and 0=Nk+v,
wherein k.sub.0 represents the GRAPPA reconstruction, w and v are
random noise terms, and k is vectorized local k-space data.
15. The tangible computer-readable medium of claim 14, wherein
calculating the k-space of the SC-GRAPPA is performed in accordance
with: k=k.sub.0-RN.sup.T(Q+NRN.sup.T).sup.-1Nk.sub.0, wherein Q and
R are covariance matrices of w and v, respectively.
16. The tangible computer-readable medium of claim 11, wherein the
k-space of the SC-GRAPPA is performed using a least squares problem
with prior estimation.
17. The tangible computer-readable medium of claim 16, wherein the
prior estimation is k.sub.0.
18. The tangible computer-readable medium of claim 11, further
comprising applying a channel combination using either the
sum-of-squares or B1-weighted summation.
19. An apparatus for determining a Self-Constraint (SC) GeneRalized
Autocalibrating Partially Parallel Acquisition (GRAPPA)
reconstruction, comprising: a memory; a processor executing
computer executable instructions in the memory, the computer
executable instructions, wherein the apparatus samples k-space data
on a set of ACS lines acquired from an MRI apparatus, estimates a
GRAPPA kernel from the ACS lines, estimates a null projection
matrix N from the ACS lines, performs a GRAPPA reconstruction to
calculate k.sub.0, and calculates a k-space of the SC-GRAPPA in
accordance with the GRAPPA reconstruction and a self-constraint
condition defined by local k-space correlations among acquired and
unacquired k-space samples.
20. The apparatus of claim 19, wherein a k-space of the SC-GRAPPA
is calculated by solving a set of linear equations in accordance
with correlations within missing k-space data.
Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application claims priority to U.S. Provisional Patent
Application No. 61/635,410, filed Apr. 19, 2012, entitled
"SELF-CONSTRAINT NON-ITERATIVE GRAPPA RECONSTRUCTION WITH
CLOSED-FORM SOLUTION," which is incorporated herein by reference in
its entirety.
BACKGROUND
[0002] Parallel magnetic resonance imaging (pMRI) techniques reduce
scan time by undersampling k-space; this directly improves temporal
resolution in real-time cine imaging and other applications.
GeneRalized Autocalibrating Partially Parallel Acquisition
(GRAPPA), a k-space based pMRI technique, is widely used clinically
for magnetic resonance imaging (MRI) because of its robustness. It
estimates the missing k-space points by solving a set of linear
equations; however, this method does not take advantage of the
correlations within the missing k-space data. In reality, all
k-space data within a neighborhood are correlated. These
correlations can be formulated as additional self-constraint
conditions, which are not considered in standard GRAPPA. While it
has been established that incorporating these self-constraints in
parallel reconstruction greatly improves the image quality, this
has only been previously demonstrated using iterative
solutions.
[0003] Conventional solutions utilize linear constraints with
iterative solvers to improve the performance of GRAPPA
reconstruction. For example, Zhao and Hu proposed to estimate a
self-constraint kernel that utilizes the reconstructed lines to
interpolate the acquired lines (iGRAPPA) (see, T. Zhao and X. Hu,
"Iterative GRAPPA (iGRAPPA) for improved parallel imaging
reconstruction," Magn Reson Med 59, 903-907 (2008)). Lustig and
Pauly proposed the SPIRIT technique, which exploits the
correlations in k-space by applying a full neighborhood kernel
(see, M. Lustig and J. M. Pauly, "SPIRiT: Iterative self-consistent
parallel imaging reconstruction from arbitrary k-space," Magn Reson
Med 64, 457-471 (2010)). Zhang et al. proposed the PRUNO technique,
which uses the null-space method to take advantage of k-space
correlations (see, J. Zhang, C. Liu and M. E. Moseley, "Parallel
reconstruction using null operations," Magn Reson Med 66, 1241-1253
(2011)).
[0004] However, these methods do not have closed-form solutions,
and the corresponding reconstruction problem can only be solved
using iterative solvers. There are some ubiquitous difficulties
associated with iterative methods. First, defining an appropriate
stopping criterion can be problematic; second, convergence may not
be guaranteed; and third, these methods are computationally
demanding and real-time reconstruction may be infeasible.
SUMMARY
[0005] Systems and methods for self-constraint GRAPPA (SC-GRAPPA)
reconstruction that provides a non-iterative, closed-form solution.
The constraints are formulated as a set of homogeneous linear
equations. The SC-GRAPPA reconstruction solves the combination of
traditional GRAPPA equations and the extra set of equations created
by the self-constraint condition. Using the simple least-squares
principle, a closed-form solution is derived without introducing
any free parameters or regularization terms. Similar to GRAPPA, the
SC-GRAPPA method can be applied using a k-space sliding window and
therefore is a direct extension of GRAPPA with
self-constraints.
[0006] In accordance with some implementations, there is provided a
method of determining a Self-Constraint (SC) GeneRalized
Autocalibrating Partially Parallel Acquisition (GRAPPA)
reconstruction. The method includes sampling k-space data on a set
of autocalibration signal (ACS) lines acquired from an MRI
apparatus; estimating a GRAPPA kernel from the ACS lines;
estimating a null projection matrix N from the ACS lines;
performing a GRAPPA reconstruction to calculate k.sub.0, which
represents the GRAPPA reconstruction; and calculating a k-space of
the SC-GRAPPA.
[0007] In accordance with some implementations, an apparatus for
determining a Self-Constraint (SC) GeneRalized Autocalibrating
Partially Parallel Acquisition (GRAPPA) reconstruction is provided.
The apparatus includes a memory and a processor executing computer
executable instructions in the memory, the computer executable
instructions. The apparatus samples k-space data from an MRI
apparatus, and part of the k-space data is a set of ACS lines,
estimates a GRAPPA kernel from the ACS lines, estimates a null
projection matrix N, performs a GRAPPA reconstruction to calculate
k.sub.0, and calculates a k-space of the SC-GRAPPA.
[0008] This summary is provided to introduce a selection of
concepts in a simplified form that are further described below in
the detailed description. This summary is not intended to identify
key features or essential features of the claimed subject matter,
nor is it intended to be used as an aid in determining the scope of
the claimed subject matter.
BRIEF DESCRIPTION OF THE DRAWINGS
[0009] The components in the drawings are not necessarily to scale
relative to each other. Like reference numerals designate
corresponding parts throughout the several views.
[0010] FIG. 1 illustrates a block diagram of an example MRI data
processing system;
[0011] FIG. 2 illustrates an example operational flow for the
computation of the SC-GRAPPA reconstruction;
[0012] FIG. 3 shows the RMSE vs. iterations of CG-PRUNO in a
phantom with acceleration rate R=6;
[0013] FIG. 4 shows the GRAPPA and SC-GRAPPA reconstructions of
data acquired in the phantom and the difference image;
[0014] FIG. 5a shows the SC-GRAPPA RMSE vs. null-space selection at
accelerate rate R=6;
[0015] FIG. 5b shows the scaled SC-GRAPPA RMSE vs. acceleration
rate;
[0016] FIGS. 5c and 5d show the results for the conventional GRAPPA
acquisition pattern. The RMSE improvement was less sensitive to
null-space selection;
[0017] FIG. 6 shows the GRAPPA reconstruction, the SC-GRAPPA
reconstruction, and 10 times the difference image in a typical
4-chamber view slice with acceleration rate R=6; and
[0018] FIG. 7 shows a typical plot of the artifact score of GRAPPA
reconstruction vs. the artifact score of SC-GRAPPA reconstruction
in the short-axis view with acceleration rate R=6.
DETAILED DESCRIPTION
[0019] The present disclosure provides example implementation for
Parallel MRI (pMRI) reconstruction techniques that reduce scan time
by undersampling k-space that estimate the missing k-space points
by solving a set of linear equations by taking advantage of the
correlations within the missing k-space data. As all k-space data
within a neighborhood are correlated, these correlations can be
estimated and regarded as self-constraint conditions which are not
considered in GRAPPA. Disclosed herein is a process that
incorporates these self-constraints in a parallel reconstruction,
without requiring an iterative solver.
[0020] In particular, the present disclosure provides an approach
to self-constraint GRAPPA (SC-GRAPPA) reconstruction that provides
a non-iterative, closed-form solution. The constraints are
formulated as a set of homogeneous linear equations. The SC-GRAPPA
reconstruction solves the combination of traditional GRAPPA
equations and an extra set of equations created by the
self-constraint condition. Using the simple least-squares
principle, a closed-form solution is derived without introducing
any free parameters or regularization terms. Similar to GRAPPA, the
SC-GRAPPA method can be applied using a k-space sliding window and
therefore is a direct extension of GRAPPA with
self-constraints.
[0021] With reference to FIG. 1, there is illustrated a block
diagram of an MRI data processing system 100 is shown in accordance
with an exemplary implementation. MRI data processing system 100
may include a magnetic resonance imaging (MRI) apparatus 101 and a
computing device 102. Computing device 102 may include a display
104, an input interface 106, a memory 108, a processor 110, and an
image data processing application 112. In the embodiment
illustrated in FIG. 1, MRI machine 101 generates MRI image
data.
[0022] Computing device 102 may be a computer of any form factor.
Different and additional components may be incorporated into
computing device 102. Components of MRI data processing system 100
may be positioned in a single location, a single facility, and/or
may be remote from one another. As a result, computing device 102
may also include a communication interface, which provides an
interface for receiving and transmitting data between devices using
various protocols, transmission technologies, and media as known to
those skilled in the art. The communication interface may support
communication using various transmission media that may be wired or
wireless.
[0023] Display 104 presents information to a user of computing
device 102 as known to those skilled in the art. For example,
display 104 may be a thin film transistor display, a light emitting
diode display, a liquid crystal display, or any of a variety of
different displays known to those skilled in the art now or in the
future.
[0024] Input interface 106 provides an interface for receiving
information from the user for entry into computing device 102 as
known to those skilled in the art. Input interface 106 may use
various input technologies including, but not limited to, a
keyboard, a pen and touch screen, a mouse, a track ball, a touch
screen, a keypad, one or more buttons, etc. to allow the user to
enter information into computing device 102 or to make selections
presented in a user interface displayed on display 104. Input
interface 106 may provide both an input and an output interface.
For example, a touch screen both allows user input and presents
output to the user.
[0025] Memory 108 is an electronic holding place or storage for
information so that the information can be accessed by processor
110 as known to those skilled in the art. Computing device 102 may
have one or more memories that use the same or a different memory
technology. Memory technologies include, but are not limited to,
any type of RAM, any type of ROM, any type of flash memory, etc.
Computing device 102 also may have one or more drives that support
the loading of a memory media such as a compact disk or digital
video disk.
[0026] Processor 110 executes instructions as known to those
skilled in the art. The instructions may be carried out by a
special purpose computer, logic circuits, or hardware circuits.
Thus, processor 110 may be implemented in hardware, firmware,
software, or any combination of these methods. The term "execution"
is the process of running an application or the carrying out of the
operation called for by an instruction. The instructions may be
written using one or more programming language, scripting language,
assembly language, etc. Processor 110 executes an instruction,
meaning that it performs the operations called for by that
instruction. Processor 110 operably couples with display 104, with
input interface 106, with memory 108, and with the communication
interface to receive, to send, and to process information.
Processor 110 may retrieve a set of instructions from a permanent
memory device and copy the instructions in an executable form to a
temporary memory device that is generally some form of RAM.
Computing device 102 may include a plurality of processors that use
the same or a different processing technology.
[0027] Image data processing application 112 performs operations
associated with performing parallel image reconstruction of
undersampled image data in k-space. Some or all of the operations
subsequently described may be embodied in image data processing
application 112. The operations may be implemented using hardware,
firmware, software, or any combination of these methods. With
reference to the exemplary embodiment of FIG. 1, image data
processing application 112 is implemented in software stored in
memory 108 and accessible by processor 110 for execution of the
instructions that embody the operations of image data processing
application 112. Image data processing application 112 may be
written using one or more programming languages, assembly
languages, scripting languages, etc.
[0028] MRI machine 101 and computing device 102 may be integrated
into a single system such as an MRI machine. MRI machine 101 and
computing device 102 may be connected directly. For example, MRI
machine 101 may connect to computing device 102 using a cable for
transmitting information between MRI machine 101 and computing
device 102. MRI machine 101 may connect to computing device 102
using a network. MRI images may be stored electronically and
accessed using computing device 102. MRI machine 101 and computing
device 102 may not be connected. Instead, the MRI data acquired
using MRI machine 101 may be manually provided to computing device
102. For example, the MRI data may be stored on electronic media
such as a CD or a DVD. After receiving the MRI data, computing
device 102 may initiate processing of the set of images that
comprise an MRI study.
[0029] With the introduction above and example operating
environment, below is an introduction to the self-constraint GRAPPA
(SC-GRAPPA) reconstruction of the present disclosure that provides
a non-iterative, closed-form solution.
[0030] Background of GRAPPA Reconstruction
[0031] The GRAPPA technique is a k-space channel-by-channel
reconstruction method. The unacquired k-space data of each channel
are estimated by interpolating the acquired k-space data of all
channels within the neighborhood. Therefore, GRAPPA can be
performed using convolution in k-space or can be computed
efficiently in the image domain using pixel-wise multiplication.
The convolution kernel can be estimated from a set of fully sampled
k-space lines, called automatic calibration signal (ACS) lines. The
GRAPPA method uses a relatively small convolution kernel which is
estimated using linear regression instead of computing directly
from the sensitivity maps.
[0032] It is well-accepted that GRAPPA is more robust than the
Sensitivity Encoding (SENSE) technique; GRAPPA can tolerate a
relatively large motion mismatch between the ACS lines and the
down-sampled k-space lines without generating severe artifacts.
While GRAPPA reconstruction is widely used in real-time cardiac
imaging because GRAPPA is robust to inaccurate ACS lines, it has
one shortcoming: GRAPPA has lower signal-to-noise ratio (SNR) than
SENSE reconstruction. In particular, it has been observed that
SENSE provides better SNR when the coil sensitivity map is accurate
and therefore shows slightly better image quality than GRAPPA. When
the coil sensitivity map has error, severe ghosting artifacts
appear in SENSE reconstruction.
[0033] GRAPPA reconstruction uses the acquired k-space data as the
only input, and the output is an estimate of the missing k-space
data. For an arbitrary k-space neighborhood, the GRAPPA technique
can be represented by the following equation:
k.sub.unacq=gk.sub.acq, (1.a)
where k.sub.unacq and k.sub.acq are vectorized unacquired and
acquired k-space data, respectively, g is the GRAPPA kernel, and
represents convolution. Since the convolution operation can be
represented as a matrix multiplication, Eq. (1) can be written as a
matrix equation:
[ 0 0 G 21 - I ] [ k acq k unacq ] = [ 0 0 ] , ( 1. b )
##EQU00001##
where G.sub.21 is the GRAPPA kernel convolution matrix, and I is
the identity matrix. Each element of Eq. (1.b) is a block.
[0034] K-Space Self-Constraint Condition
[0035] All k-space data have local correlations. These correlations
can be represented by a set of extra equations that are not
included in standard GRAPPA (1). The method of the present
disclosure incorporates these equations to generate a new (larger)
set of equations. Image reconstruction may be performed by finding
the optimal solution to the new set of equations. Three recently
published papers proposed different approaches to assemble the
equations which originate from the local k-space correlations. In
the following sections, the three techniques are reviewed, followed
by a description of the self-constraint GRAPPA.
[0036] Iterative GRAPPA (iGRAPPA) is a method that uses missing
data to interpolate the acquired data as a self-constraint
condition that can be summarized using the following matrix
equations:
[ k acq k unacq ] = [ 0 G 12 G 21 0 ] [ k acq k unacq ] ( 2 )
##EQU00002##
where G.sub.21 is the GRAPPA kernel convolution matrix (identical
to Eq. (1.b)), and the G.sub.12 is the matrix that represents the
self-constraint condition.
[0037] SPIRiT (iterative self-consistent parallel imaging
reconstruction) utilizes a full-neighborhood kernel to interpolate
the unacquired k-space data. Mathematically, SPIRiT can also be
written as a self-constraint condition where each sample in
k-space, acquired or unacquired, can be represented by a weighted
sum of neighboring acquired and unacquired samples, yielding:
[ k acq k unacq ] = [ S 11 S 12 S 21 S 22 ] [ k acq k unacq ] or (
3. a ) [ S 11 - I S 12 S 21 S 22 - I ] [ k acq k unacq ] = [ 0 0 ]
( 3. b ) ##EQU00003##
[0038] The matrix S is the SPIRiT convolution kernel. Eq. (1) and
Eq. (2) can be considered as special cases of Eq. (3). The matrices
G and S in Eq. (2) and Eq. (3), respectively, can both be estimated
using linear regression of the ACS lines, similar to the GRAPPA
kernel estimation procedure.
[0039] Another method that utilizes k-space correlations is PRUNO
(parallel reconstruction using null operations). In PRUNO
formulation, every k-space neighborhood can be projected onto a
null space by the same projection matrix N:
[ N 11 N 12 N 21 N 22 ] [ k acq k unacq ] = [ 0 0 ] ( 4 )
##EQU00004##
Instead of the linear regression method, PRUNO uses singular value
decomposition (SVD) to estimate the projection matrix N.
[0040] The similarity of Eq. (2), Eq. (3) and Eq. (4) indicates how
all three methods take advantage of the local k-space correlations
among acquired and unacquired samples that GRAPPA does not. Both
SPIRIT and PRUNO use full k-space kernels, and iGRAPPA uses kernels
that either span the acquired or the unacquired k-space lines. In
other words, SPIRIT and PRUNO are able to exploit more generic
correlations than iGRAPPA. However, both SPIRIT and PRUNO utilize
iterative solvers for image reconstruction.
[0041] There are differences, however, between SPIRIT and PRUNO.
First, two different mathematical tools are used to solve these
equations, i.e., linear regression for SPIRIT vs. SVD for PRUNO.
Second, the number of equations is different. The number of rows of
Eq. (4) is higher than that of Eq. (3); PRUNO results in more
equations because it utilizes the null-space kernels to interpolate
k-space. The number of null-space kernels can be considerably
larger than the number of channels. In contrast, the SC-GRAPPA of
the present disclosure uses GRAPPA reconstruction as a prior and
uses PRUNO equations as the self-constraint condition.
[0042] Self-Constraint GRAPPA (SC-GRAPPA) Reconstruction
[0043] Unlike iGRAPPA/SPIRIT/PRUNO, the SC-GRAPPA method has a
closed-form solution. There are two mathematical methods that can
be used to formulate this problem; the first is a convex
optimization with linear constraints, and the second is the least
squares (LS) problem with prior estimation. It can be proven that
the second approach is more generic.
[0044] The linear equations of SC-GRAPPA are as follows:
k=k.sub.0+w (5.a)
0=Nk+v (5.b)
where k.sub.0 in Eq. (5.a) represents the GRAPPA reconstruction,
used as the prior estimation. Both k and k.sub.0 are vectorized
local k-space data. Eq. (5.b) is the self-constraint condition from
PRUNO. The definition of matrix N is the same as Eq. (4). Both w, v
are random noise terms with covariance matrices Q and R,
respectively.
[0045] From Eq. (5), there are two intuitive ways to formulate the
self-constraint condition: it can either be formulated as a problem
of convex optimization with linear constraints, or as a least
squares problem with prior estimation. In the following section, we
will manifest both formulations and reveal the relation between
them.
[0046] The corresponding convex optimization problem with linear
constraint:
min.sub.x1/2.parallel.k-k.sub.0.parallel..sup.2s.t.Nk=b (A.1)
where k, k.sub.0, b and N are vectors and matrix in the complex
domain. N.sup.H represents the Hermitian transpose of matrix N. Eq.
(A.1) can be solved using the Lagrange method, generating the
following linear equations:
[ I - N H - N 0 ] [ k .lamda. ] = [ k 0 - b ] ( A .2 )
##EQU00005##
with solution:
[ k .lamda. ] = [ ( 1 - N H ( NN H ) - 1 N ) k 0 + N H ( NN H ) - 1
b - ( NN H ) - 1 Hk 0 + ( NN H ) - 1 b ] , ( A .3 )
##EQU00006##
which can be written as:
k=k.sub.0+N.sup.H(NN.sup.H).sup.-1(b-Nk.sub.0) (A.4)
[0047] The alternative formulation is the least-squares problem
with prior estimation:
k.sub.0=k+w
b=Nk+v, (A.5)
where w and v are random noise terms, representing the
uncertainties in both equations. In matrix form:
[ b k 0 ] = [ N I ] k + [ v w ] . ( A .6 ) ##EQU00007##
[0048] Assuming that the noise covariance matrix can be written
as:
P = [ Q 0 0 R ] . ( A .7 ) ##EQU00008##
A linear unbiased estimation is:
k = ( N H Q - 1 N + R - 1 ) - 1 ( N H Q - 1 b + R - 1 k 0 ) = k 0 +
RN H ( Q + NRN H ) - 1 ( b - Nk 0 ) . ( A .8 ) ##EQU00009##
[0049] The second formulation can be converted back to the first
formulation under proper assumptions, if there are two
pre-conditions:
(I+NN.sup.H).sup.-1=(NN.sup.H).sup.-1((NN.sup.H).sup.-1+I).sup.-1.apprxe-
q.(NN.sup.H).sup.-1, (A.9.a)
Q=R=I (A.9.b)
then Eq. (A.8) can be rewritten as:
k=k.sub.0+N.sup.H(NN.sup.H).sup.-1(b-Nk.sub.0) (A.10)
which is exactly the same as Eq. (A.4). Eq. (A.8) is a more general
form of Eq. (A.4).
[0050] Thus, according to Eq. (A.5) to Eq. (A.8), the LS solution
of Eq. (5) is:
k=k.sub.0-RN.sup.T(Q+NRN.sup.T).sup.-1Nk.sub.0 (6)
[0051] To estimate the covariance matrices Q and R, these may be
set as the identity matrix for purposes of simplicity. Eq. (6) can
be implemented as a convolution in k-space, and can be calculated
in every local k-space neighborhood using a sliding window. Since
each k-space data point can be assessed multiple times when a
sliding window is applied, a simple average is used to reach the
final result. For example, if a 5.times.5 sliding window is
selected, and an 8-channel phase-array coil is used, then the input
k.sub.0 in Eq. (6) is a 1-D vector with dimension 1.times.200, and
the output k is a 1-D vector of the same size. If the full k-space
dimension is 128.times.128, then the k-space data can be vectorized
as 16384 1-D vectors, each with dimension 1.times.200. Because each
k-space point appears in exactly 25 vectors, and each may have a
different value, the final result will be the average of these 25
values.
[0052] The SC-GRAPPA results in higher noise suppression in image
regions with lower SNR. In the high SNR regions, the signal content
usually satisfies the self-constraint condition automatically, and
applying the additional self-constraint condition may not lead to
significant improvement. In the low SNR regions, the random noise
does not satisfy the self-constraint condition, and applying the
additional self-constraint condition shows stronger noise reduction
effect. This property may help improve the local image SNR in
regions where signal penetration is problematic.
[0053] There is a generic trade-off between the artifact level and
the SNR. Typically, any improvement in SNR comes with a
corresponding increase in artifact level. As detailed below in a
phantom study, more aggressive selection of the null-space
dimension lead to higher aliasing/artifacts, but lower random
noise. In-vivo experiments demonstrated that SC-GRAPPA improves
both SNR and artifact over GRAPPA when the null-space selection was
simply set to the number of phased-array coil channels.
[0054] Thus, optimal selection of the sliding window size and the
null-space dimension should improve the performance of SC-GRAPPA.
However, there are no well-established methods to optimally select
either parameter. The same problem also exists in the SPIRiT and
PRUNO techniques. As such, the SC-GRAPPA method of the present
disclosure depends on the sliding window size and null-space
dimension of the PRUNO method, but is not sensitive to either
parameter. In fact, SC-GRAPPA showed significant improvement over
GRAPPA for all three sliding window sizes: 7.times.7, 9.times.9 and
11.times.11. In addition, the RMSE vs. the selection of the
null-space dimension curve is rather flat. This indicates that even
a sub-optimal null-space dimension selection in SC-GRAPPA would
still have lower RMSE and higher SNR than GRAPPA. Therefore, a
relatively conservative null-space selection and a relatively large
sliding window may be adopted to provide an improvement over GRAPPA
reconstruction.
[0055] The performance of SC-GRAPPA can be further improved by
incorporating additional data to fine tune the parameters. For
example, if two sets of ACS lines are available, the
cross-validation method can be used to find the optimal sliding
window size and the modes of the null-space selection; one set of
ACS lines can be used for calibration and the other set of ACS
lines can be used for reconstruction validation. The optimal
parameters can be determined by finding the minimal residual. As a
practical approach, several pre-scan low-resolution images with
variant slice location/orientation may also be used to further
optimize the parameters.
[0056] With reference to FIG. 2, there is illustrated an example
operational flow 200 for the computation of the SC-GRAPPA
reconstruction. At 202, the GRAPPA kernel is estimated from ACS
lines. At 204, the null projection matrix N is estimated from ACS
lines. At 206, the GRAPPA reconstruction is performed to calculate
k.sub.0. At 208, a k-space of SC-GRAPPA is calculated. For example,
Eq. (6) above may be used to calculate the k-space. At 210, the
channel combination is applied using either the sum-of-squares or
B1-weighted summation.
[0057] In a prototype SC-GRAPPA implementation, the reconstruction
was performed either in k-space, or in image space. Experiments
showed that the computational time was relatively low by
implementing the k-space convolution as an image space point-wise
multiplication. Thus, in summary, the present disclosure presents a
new framework to combine the self-constraint condition with the
GRAPPA technique to improve the reconstructed image quality using a
closed-form solution. This proposed framework combines GRAPPA with
other linear constraints given in, e.g., Eq. (5.b).
[0058] Methods
[0059] A. Phantom Study
[0060] Seventeen time frames were acquired from the coronal
orientation slice of two spherical water bottles using a
steady-state free precession (SSFP) real-time cine sequence with no
parallel acceleration. Detailed parameters for the SSFP sequence
were: flip angle=70.degree., TE=1.15 ms, TR=2.28 ms, pixel
bandwidth=1502 Hz/pixel, matrix size=128.times.120, field of view
(FOV)=400.times.375 mm.sup.2, and slice thickness=5.0 mm.
[0061] The first frame of the image series was used for the ACS
lines for GRAPPA, CG-PRUNO, and SC-GRAPPA. The remaining 16 fully
sampled frames were uniformly down-sampled and then reconstructed
using GRAPPA, CG-PRUNO, and SC-GRAPPA. All reconstructions were
tested for every possible k-space trajectory, e.g., for R=6, all
six possible uniform down-sampled k-space trajectories were
reconstructed. The kernel size used in GRAPPA reconstruction was
4.times.5; the kernel size used in SC-GRAPPA reconstruction was
7.times.7; and the sliding window sizes used in null-space
selection were 7.times.7, 9.times.9 and 11.times.11. Using the
corresponding fully sampled k-space as the gold standard, the
root-mean-square-error (RMSE) of the reconstructed k-space was
evaluated as a metric to measure the fidelity of both
reconstructions. Since there is no well-established method to find
the optimal dimension of the null-space, we plotted the RMSE vs.
null-space dimension selection to study it qualitatively. Both
GRAPPA and SC-GRAPPA reconstructions and the corresponding RMSE
measurement were repeated using the conventional GRAPPA acquisition
pattern. Forty ACS lines in the center of k-space were fully
sampled, and the rest of k-space was uniformly down-sampled (please
note that this strategy reduces the effective acceleration
rate).
[0062] B. In-Vivo Real-Time Cardiac MR Cine Images
[0063] SC-GRAPPA was tested in multiple cardiac views in two
healthy volunteers. MR real-time cardiac cine images with uniformly
down-sampled temporally-interleaved k-space were reconstructed
using both GRAPPA and SC-GRAPPA. We acquired two SSFP real-time
cine series using TGRAPPA with parallel acceleration rates R=5 and
R=6, in vertical and horizontal long-axis, and one short-axis
views. Imaging parameters (for R=5/6) were 192.times.95/96 matrix
reconstructed from 192.times.15/16 acquired matrix, 6 mm thick
slice, flip angle=48.degree., TE/TR=1.0/2.56 ms, pixel
bandwidth=1447 Hz/pixel, FOV=380.times.285 mm.sup.2. A total of 256
frames were acquired per image series to support statistical
analysis. The ACS lines were generated using average-all-frames. A
7.times.7 sliding window size was used in the volunteer study. The
null-space selection was set to be the same as the number of
channels, i.e. only 32 modes belong to signal-space, and all other
modes belong to null-space.
[0064] After reconstruction, the signal was estimated as the
temporal mean. The noise level of images was evaluated using the
MP-law method based on the random matrix theory. The Karhunen-Loeve
transform was applied along the temporal direction, and the
noise-only eigenimages were identified by assuming they follow a
specific distribution function, i.e., the Marcenko-Pastur law. Once
identified, the noise level can be accurately estimated from the
noise-only eigenimages. The image artifact level was estimated
using a spatial cross-correlation based method. A template
including 1% of the highest signal pixels was generated for each
frame, typically comprising the brightest subcutaneous fat of the
chest wall. An artifact score was defined as the cross-correlation
coefficient evaluated between the template and the corresponding
frame at phase encoding FOV/R (where R is the acceleration
rate).
[0065] In the studies described below, phantom and human images
were acquired on a 1.5T MRI system (MAGNETOM Avanto, Siemens
Healthcare Inc., Erlangen, Germany). A 32-channel cardiac array
coil (Quality Electrodynamics, Mayfield Village, Ohio) was used for
data acquisition. The SC-GRAPPA method was implemented and all
images processed using Matlab.RTM. 2011a (Math Works, Natick,
Mass.) running on a computer with Intel.RTM. Xeon.RTM. E5620 2.4
GHz CPU, 40 GB system RAM. Utilizing Eq. (6), the SC-GRAPPA
reconstruction can be computed in k-space using a sliding window.
The convolution operation in Eq. (6) was implemented in image space
as a pixel-wise multiplication to avoid the time-consuming k-space
convolution and thereby increase the speed of reconstruction
significantly.sup.6. This image space implementation is a direct
application of the convolution theorem. The conjugate gradient
(CG)-PRUNO algorithm without regularization terms was also
implemented in Matlab using a 7.times.7 kernel and applied in the
phantom study to provide a comparison with another self-constraint
reconstruction method.
[0066] IV Results
[0067] A. Phantom Study
[0068] FIG. 3 shows the RMSE vs. iterations of CG-PRUNO in a
phantom with acceleration rate R=6. For CG-PRUNO reconstruction,
the RMSE is consistently worse than GRAPPA reconstruction,
regardless of the null-space selection. In addition, large
fluctuation was observed among different k-space trajectories
(>20%). However, the SC-GRAPPA reconstruction shows a consistent
RMSE improvement over GRAPPA reconstruction. The relative RMSE
fluctuation for GRAPPA reconstruction in all six trajectories is
less than 1.1%, and for SC-GRAPPA is less than 1.2% for all
null-space selections. The relative RMSE fluctuation is even
smaller at lower acceleration rate. The reconstruction times for
GRAPPA and SC-GRAPPA are 0.19 s and 2.0 s, respectively. The
reconstruction time for CG-PRUNO varies from 2.0 s to 15 s,
depending on the k-space trajectory.
[0069] FIG. 4 shows the GRAPPA and SC-GRAPPA reconstructions of
data acquired in the phantom, and the difference image. The
acceleration rate R=6, the sliding window size was 11.times.11, and
the null-space selection was 48. Note that the noise reduction in
the air region was more significant than in regions of the image
with high signal.
[0070] FIG. 5a shows the SC-GRAPPA RMSE vs. null-space selection at
accelerate rate R=6. When the null-space was properly selected, the
RMSE of SC-GRAPPA was smaller than GRAPPA. FIG. 5b shows the scaled
SC-GRAPPA RMSE vs. acceleration rate. The RMSE of GRAPPA
reconstruction was set to be 100% for all acceleration rates.
SC-GRAPPA shows higher RMSE reduction at high acceleration. FIGS.
5c and 5d show the results for the conventional GRAPPA acquisition
pattern. The RMSE improvement was less sensitive to null-space
selection. The same results were observed for all three sliding
window sizes; 7.times.7, 9.times.9, and 11.times.11. The maximum
RMSE reduction was consistently higher when the sliding window was
larger. The optimal PRUNO mode selection also increased with the
sliding window size.
[0071] B. In-Vivo Real-Time Cardiac MR Cine Images
[0072] FIG. 6 shows the GRAPPA reconstruction, the SC-GRAPPA
reconstruction, and 10 times the difference image in a typical
4-chamber view slice with acceleration rate R=6. The difference
image shows that the SNR improvement of SC-GRAPPA was not spatially
homogeneous. Low SNR regions of the image showed the highest noise
reduction. In the lung region marked in FIG. 6, the noise variance
was reduced by 54%, which is more than the effect of averaging two
acquisitions. In high SNR regions, e.g. in the chest wall, the
SC-GRAPPA reconstruction and the GRAPPA reconstruction were nearly
identical. The difference image showed spatially variant noise.
[0073] FIG. 7 shows a typical plot of the artifact score of GRAPPA
reconstruction vs. the artifact score of SC-GRAPPA reconstruction
in the short-axis view with acceleration rate R=6. The SC-GRAPPA
reconstruction showed significantly lower artifact level
(p-value<0.001). At both acceleration rates (R=5 and R=6), the
Student t-test showed that the SC-GRAPPA reconstruction had a lower
artifacts level than the GRAPPA reconstruction
(p-value<0.001).
[0074] Table I, below, shows the global SNR gain of SC-GRAPPA over
GRAPPA. The SNR improvements are statistically significant
(p<0.001) for both acceleration rates R=5 and R=6.
TABLE-US-00001 Slice 1 Slice 2 Slice 3 Mean Volunteer 1 Rate 5
12.6% 10.9% 21.9% 15.13% (p < 0.001) Rate 6 9.0% 3.3% 18.0%
10.10% (p < 0.001) Volunteer 2 Rate 5 9.8% 14.6% 18.9% 14.43% (p
< 0.001) Rate 6 14.2% 9.4% 5.0% 9.53% (p < 0.001)
[0075] Thus, the present disclosure describes a new
channel-by-channel k-space image reconstruction method by
incorporating the self-constraint condition into the standard
GRAPPA reconstruction method. The SC-GRAPPA framework has a
closed-form solution and can be applied whenever GRAPPA can be
applied. The technique has overall higher SNR than GRAPPA in
real-time dynamic cardiac cine imaging, with higher gain in regions
of low SNR. SC-GRAPPA offers improved SNR over GRAPPA and should
provide advantages in situations where high acceleration rates are
needed.
[0076] It should be understood that the various techniques
described herein may be implemented in connection with hardware or
software or, where appropriate, with a combination of both. Thus,
the methods and apparatus of the presently disclosed subject
matter, or certain aspects or portions thereof, may take the form
of program code (i.e., instructions) embodied in tangible media,
such as floppy diskettes, CD-ROMs, hard drives, or any other
machine-readable storage medium wherein, when the program code is
loaded into and executed by a machine, such as a computer, the
machine becomes an apparatus for practicing the presently disclosed
subject matter. In the case of program code execution on
programmable computers, the computing device generally includes a
processor, a storage medium readable by the processor (including
volatile and non-volatile memory and/or storage elements), at least
one input device, and at least one output device. One or more
programs may implement or utilize the processes described in
connection with the presently disclosed subject matter, e.g.,
through the use of an API, reusable controls, or the like. Such
programs are preferably implemented in a high level procedural or
object oriented programming language to communicate with a computer
system. However, the program(s) can be implemented in assembly or
machine language, if desired. In any case, the language may be a
compiled or interpreted language, and combined with hardware
implementations.
[0077] Although example embodiments may refer to utilizing aspects
of the presently disclosed subject matter in the context of one or
more stand-alone computer systems, the subject matter is not so
limited, but rather may be implemented in connection with any
computing environment, such as a network or distributed computing
environment. Still further, aspects of the presently disclosed
subject matter may be implemented in or across a plurality of
processing chips or devices, and storage may similarly be effected
across a plurality of devices. Such devices might include personal
computers, network servers, and handheld devices, for example.
[0078] Although the subject matter has been described in language
specific to structural features and/or methodological acts, it is
to be understood that the subject matter defined in the appended
claims is not necessarily limited to the specific features or acts
described previously. Rather, the specific features and acts
described previously are disclosed as example forms of implementing
the claims.
* * * * *